Library organizations in the Netherlands show an increasing interest to employ depots for low-cost storage and demand fulfillment of item requests. Typically, all libraries in an organization have a shared catalog, and, on local unavailability, requests can be shipped from elsewhere in the organization. The depot can be used to consolidate shipment requests by making tours along all libraries, delivering requested items, but also picking up items that have to be stored at the depot, or that have to be shipped from one library to another. Cross docking and delayed shipments are two preferred methods for fulfilling requests that cannot be directly met using on-hand stock at the depot. In this paper, we compare these two methods from an inventory control perspective. We model the library system as a Markov Decision process. For one- and two-location systems, we derive analytical results for the average-cost optimal policy, showing that the decision to store items from the location at the depot satisfies a threshold structure depending on the number of rented items. For larger instances, an effective heuristic is proposed exploiting this threshold structure. In numerical experiments, important managerial insights are obtained by comparing cross docking and delayed shipments in different situations. Cross docking is shown to add most value in systems with low total stock, however, delayed shipments may achieve similar costs as cross docking when stock is high or when tours frequently visit all locations. Furthermore, effective decisions can be based on simple model formulations with memoryless rental time distributions.